Local Polynomial Estimation in Irregular Correlated Random Coefficient Panel Data Models

We study estimation of the average partial effect (APE) in an irregular correlated random coefficients panel data model in which the time dimension equals the number of random coefficients, $T=p$, and the density function is degenerate. We extend Graham and Powell (2012) in two directions. First, we allow the density of $d_i=\det(X_i)$ to vanish or diverge at the origin at the polynomial rate $|d|^{a}$ with $a>-1$, and show that the aggregate time effects and the APE remain identified for every such $a$, extending earlier work studying the case $a = 0$. Second, we propose a three-step estimator: an $m$th-order local polynomial regression over the stayers for the time effects; a local polynomial fit of the conditional mean coefficient just outside the trimming threshold; and an imputation step that extrapolates the latter into the stayer region rather than discarding stayers. We establish joint asymptotic normality in three regimes determined by the sign of $a$, and obtain a convergence rate approaching $n^{-(m+1)/(2m+3)}$ when $a=0$, against $n^{-1/3}$ for Graham and Powell (2012), whose estimator is the special case $m=0$, $a=0$. The results show that smoothness of the underlying conditional expectations can be traded for a faster convergence rate in irregularly identified models. Simulations confirm lower bias and root mean squared error across designs. We redo the overidentification analysis of a markup model in Raval (2023), allowing for correlated random elasticities.